On 7 Dez., 21:18, William Hughes <wpihug...@gmail.com> wrote:
> Your claim is that any increasing sequence of sets > must contain its limit.
No. On the contrary I have explicitly excluded the limit.
> This is trivially equivalent to > any increasing sequence of sets must have a last element.
Please refrain from your pure conjectures. Please focus on mathematical arguments: There is an inclusion-monotonic sequence without its limit. > > If we combine this with "there is no largest natural number" > then we immediately get "there is no set containing all > natural numbers". Hence your poorly defined concept > "potential infinity".
That is by no means poorly defined. It is not definable in set theory. But that does not matter. Here are some English sources:
"Cantor's work was well received by some of the prominent mathematicians of his day, such as Richard Dedekind. But his willingness to regard infinite sets as objects to be treated in much the same way as finite sets was bitterly attacked by others, particularly Kronecker. There was no objection to a 'potential infinity' in the form of an unending process, but an 'actual infinity' in the form of a completed infinite set was harder to accept."
Fraenkel, Abraham A., Levy, Azriel: "Abstract Set Theory" (1976): "the set of all integers is infinite (infinitely comprehensive) in a sense which is "actual" (proper) and not "potential". (p.6) One may doubt whether this example really illustrates the abyss between finiteness and actual infinity.(p.6)
Thomas Jech, Set Theory Stanford.htm, Stanford Encyclopedia of Philosophy: Until then, no one envisioned the possibility that infinities come in different sizes, and moreover, mathematicians had no use for ?actual infinity.? The arguments using infinity, including the Differential Calculus of Newton and Leibniz, do not require the use of infinite sets, > > However, you also claim that it is inconsistent to assume > that there can be an increasing sequence of sets without > last element.
Not at all! That incresing sequence without a (fixed) last element ist the only possible way to understand the natural numbers.
> You have umpteen justifications for this, all > of which are circular as they always rely sooner or later > on "any increasing sequence of sets must have a last element"
Please try to think purely formally: There is a sequence with only finite sets. This sequence contains all natural numbers (by definition of sequence). There are not two sets let alone infinitely many sets that are containing more than one of them. By the way I could never understand how you dared to argue already three years ago, that the union of FISONs could contain more than each of the FISIONs. But that is no longer a problem as we now refrain from any unioning at all.
